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Albert algebras over rings and related torsors

Part of: Jordan algebras Algebraic groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  30 March 2020

Seidon Alsaody
Seidon Alsaody*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, ABT6G 2G1, Canada
*
e-mail: [email protected]
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Abstract

We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures.

We begin by showing that isotopes of Albert algebras are obtained as twists by a certain $\mathrm F_4$ -torsor with total space a group of type $\mathrm E_6$ and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain $\mathrm D_4$ -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

Keywords

Albert algebraJordan algebrahomogeneous spacetorsorisotopecompositionoctonion

MSC classification

Primary: 17C40: Exceptional Jordan structures
Secondary: 20G41: Exceptional groups 14L30: Group actions on varieties or schemes (quotients) 20G10: Cohomology theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 3 , June 2021 , pp. 875 - 898
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

An extensive part of this work was done while the author was a postdoctoral fellow at Institut Camille Jordan (Lyon, France), supported by the grant KAW 2015.0367 from the Knut and Alice Wallenberg Foundation. Part of this work was done while the author was partially supported by PIMS at the University of Alberta.

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